Blender 2.8 - Transparent background
Transparent Background
In general - Render Properties - Film - Transparent works (viewport and render)

To Save render with transparency - Render Properties - Output - File Format - PNG and Color - RGBA has to be set (or any other format that supports alpha channel).

Hide Background
For viewport you can hide background image in Shading properties by changing World Opacity set to zero.

Notes:

Works for Eevee and Cycles
HDRi is file format, but important in terms is that background in blender use equirectangular texture type. It doesnt has to be necessarely HDRi that refers to a high dynamic range image.